It is not Alcatraz, nor Robben Island, and Roman Polanski is not singing Jailhouse Rock. The film director’s current prison, the chalet Milky Way, looks out on to his favourite ski slopes on the Eggli mountain and, even if you close the window, you can hear the church bells from down the valley.

“We miss him — he’s always cheerful,” said the owner of the bakery where Polanski used to buy his croissants. Now his electronic foot tag would sound the alarm if he left his golden cage and strayed down the hill for breakfast.

The 76-year-old film-maker, officially a fugitive from American justice, has been in and out of Switzerland for 30 years and was treated as a welcome celebrity before suddenly being arrested by the Swiss authorities in what was seen by many as an attempt to curry favour with the United States.

On Friday, a Los Angeles court will consider Polanski’s plea to be tried in absentia. If granted, it will allow his lawyers to explain in detail the reasons why he fled the US in 1978 while awaiting sentencing for the sexual abuse of a 13-year-old girl — and negate any need to extradite him.

In the meantime, warns Roger Seifriz, the head of tourism in the region, the locals should stop talking to the press about Polanski. “Is it correct to answer all journalist questions?” he asked in an open letter to the people of Gstaad. “Or would it be more appropriate to think twice and then reply: ‘I’m sorry, I can’t speak about it’?” Mr Seifriz is plainly worried not just about Polanski’s privacy, but also about scaring away Gstaad’s remaining publicity-shy celebrities. The village has got the message and has, for the most part, closed ranks. Three tribes inhabit Gstaad. The first are locals; the second, the well-off tourists, many from Russia, who ski, shop, eat expensively and then go home. The third is the chalet tribe.

“Stick around until midnight — that’s when they come out,” says Andrea Scherz, the owner of the Palace Hotel, which looms over Gstaad like Kafka’s Castle.

Sure enough, the huge lobby, with its crackling wood fire, antlers, deep, foetal chairs and backgammon boards, is the social hub of the chalet world.

“They come to see and be seen,” says Mr Scherz, whose grandfather bought the place after the war. He persuaded Louis Armstrong to play in the lobby, and brought in crooners such as Maurice Chevalier.

The most expensive chalets, which sell for as much as €7 million (£6 million), are in the Oberbort district of Gstaad, close enough to the Palace to make it into a second living room.

There the crowd swells and ebbs as in a railway terminus: the half-brother of Osama bin Laden; the Formula One boss Bernie Ecclestone (who owns a hotel on the Gstaad Promenade); the dress designer Valentino (watch out for his liveried servant walking the dogs); assorted racing drivers and, naturally, legions of beautiful young women.

Polanski admits to having schoolgirls around to the chalet after the murder of his wife, Sharon Tate, in 1969. “Kathy, Madeleine, Sylvia and others whose names I forget played a fleeting but therapeutic role in my life,” he wrote in his memoirs. “They were all between 16 and 19 years old . . .”

The chalet community is divided about him. Some dinner hostesses refuse to have his name mentioned at table, others see him as a relic of the 1970s, when nasty Los Angeles vices were imported to the valley.

In the old days, Gstaad was the hangout for British Hollywood: David Niven (who is buried just down the valley), Peter Sellars, Richard Burton, Joan Collins, even Julie Andrews. Polanski, by contrast, brought Jack Nicholson to the place: an altogether different kettle of fish.

The Gstaad super-rich are, though, mostly forgiving of the delinquent director. It is a place where people can afford to be politically incorrect.

“He works hard, plays hard,” says a backgammon player in the Palace. “They should leave him alone.”